| L(s) = 1 | + (−1.31 + 0.515i)2-s + (−0.275 + 1.70i)3-s + (1.46 − 1.35i)4-s + (0.241 − 0.820i)5-s + (−0.518 − 2.39i)6-s + (−2.26 − 1.96i)7-s + (−1.23 + 2.54i)8-s + (−2.84 − 0.942i)9-s + (0.105 + 1.20i)10-s + (−5.20 + 3.34i)11-s + (1.91 + 2.88i)12-s + (−0.429 − 0.495i)13-s + (4.00 + 1.42i)14-s + (1.33 + 0.638i)15-s + (0.314 − 3.98i)16-s + (−0.947 − 0.432i)17-s + ⋯ |
| L(s) = 1 | + (−0.931 + 0.364i)2-s + (−0.159 + 0.987i)3-s + (0.734 − 0.678i)4-s + (0.107 − 0.367i)5-s + (−0.211 − 0.977i)6-s + (−0.857 − 0.743i)7-s + (−0.436 + 0.899i)8-s + (−0.949 − 0.314i)9-s + (0.0334 + 0.381i)10-s + (−1.56 + 1.00i)11-s + (0.553 + 0.833i)12-s + (−0.119 − 0.137i)13-s + (1.06 + 0.379i)14-s + (0.345 + 0.164i)15-s + (0.0786 − 0.996i)16-s + (−0.229 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0151195 - 0.0405308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0151195 - 0.0405308i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.515i)T \) |
| 3 | \( 1 + (0.275 - 1.70i)T \) |
| 23 | \( 1 + (-0.944 + 4.70i)T \) |
| good | 5 | \( 1 + (-0.241 + 0.820i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.26 + 1.96i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (5.20 - 3.34i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.429 + 0.495i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.947 + 0.432i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (6.17 - 2.82i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (8.05 + 3.67i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-3.76 - 0.540i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.27 - 0.668i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.65 - 5.62i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.95 + 0.855i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + (-2.19 - 1.89i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.85 + 6.75i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.35 - 9.45i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.75 - 5.85i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.71 - 3.67i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.30 + 2.85i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (8.12 - 7.04i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.62 + 1.65i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (4.31 - 0.620i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.80 + 1.11i)T + (81.6 + 52.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41237181401493598069190266357, −10.93409822357494870254512335589, −10.34916674899488809785974248565, −9.788869297672249049769981851485, −8.785087575058289149622863085376, −7.76736521043911659397834235056, −6.63627802557248698336365973377, −5.48783321009461671443178327914, −4.36653791157120541351023965804, −2.60207614952861877374241246477,
0.04042617583061684009399423063, 2.28613816946731156784050387211, 3.09823959737687143365833863053, 5.63371583358373984903799082211, 6.51506163703399585273132360936, 7.48023270414577890541442957756, 8.481751034551801180512519576864, 9.207152397344207740999540166384, 10.65103543121091915000756882696, 11.04272227867106831607825469679
